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Chapter 2: Problem 124

The principal argument of \(-2 \sqrt{3}-2 \mathrm{i}\) is (a) \([(-5 \pi) / 6]\) (b) \([(5 \pi) / 6]\) (c) \([(-2 \pi) / 3]\) (d) \([(2 \pi) / 3]\)

Short Answer

Expert verified
The principal argument of the given complex number is \(-\frac{2 \pi}{3}\), which corresponds to option (c).

Step by step solution

01

Express the given complex number in rectangular form

The given complex number is -2√3 - 2i.
02

Identify the real and imaginary parts of the complex number

In the given complex number -2√3 - 2i, the real part is -2√3, and the imaginary part is -2.
03

Calculate the angle (theta) using the arctan function

The formula to find the angle of the complex number is: θ = arctan(Im(z)/Re(z)) Where Im(z) is the imaginary part, and Re(z) is the real part of the complex number. Θ = arctan((-2)/(-2√3))
04

Evaluate the arctan function

Upon evaluating the arctan function, we get: θ = arctan(1/√3) Since we want to find the principal argument of the complex number, ensure the angle lies between -π and π: θ = - π/3 ≈ -1.047
05

Find the corresponding option in terms of π

Since θ = - π/3 ≈ -1.047 , let's check which option matches this angle: (a) - (5π) / 6 (b) (5π) / 6 (c) - (2π) / 3 (d) (2π) / 3 The correct option is (c) - (2π) / 3 because it matches the calculated angle of θ = - π/3.

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Most popular questions from this chapter

Let \(z\) be complex number with modulus 2 and argument \([(-2 \pi) / 3]\) then \(\mathrm{z}=\) (a) \(-1+\mathrm{i} \sqrt{3}\) (b) \([(-1+i \sqrt{3}) / 2]\) (c) \(-1-\mathrm{i} \sqrt{3}\) (d) \([(-1-i \sqrt{3}) / 2]\)

If \(z=x+i y, x, y \in R\) and \(3 x+(3 x-y) i=4-6 i\) then \(z=\) (a) \((4 / 3)+\mathrm{i} 10\) (b) \((4 / 3)-\mathrm{i} 10\) (c) \([(-4) / 3]+\mathrm{i} 10\) (d) \([(-4) / 3]-\mathrm{i} 10\)

If \(\mathrm{a}=\cos (2 \pi / 7)+\mathrm{i} \sin (2 \pi / 7)\) then the quadratic equation whose roots are \(\alpha=\mathrm{a}+\mathrm{a}^{2}+\mathrm{a}^{4}\) and \(\beta=\mathrm{a}^{3}+\mathrm{a}^{5}+\mathrm{a}^{6}\) is (a) \(x^{2}-x+2\) (b) \(x^{2}+x-2\) (c) \(x^{2}-x-2\) (d) \(x^{2}+x+2\)

\(\mathrm{i}^{1}+\mathrm{i}^{2}+\mathrm{i}^{3}+\mathrm{i}^{4}+\ldots \ldots \ldots \mathrm{i}^{1000}=\) (a) \(-1\) (b) 0 (c) 1 (d) None

The complex numbers \(\mathrm{z}_{1}, \mathrm{z}_{2}\) and \(\mathrm{z}_{3}\) satisfying \(\left[\left(z_{1}-z_{3}\right) /\left(z_{2}-z_{3}\right)\right]=[(1-i \sqrt{3}) / 2]\) are the vertices of a triangle which is (a) of area zero (b) right angled isosceles (c) equilateral (d) obtuse-angle isosceles
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